Problem 153
Question
If \(A=\left[\begin{array}{ccc}1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{array}\right]\), then find the trace of matrix \(A\)
Step-by-Step Solution
Verified Answer
The trace of matrix A is 17.
1Step 1: Identify the elements on the main diagonal of the matrix
In the given matrix A, the elements on the main diagonal are 1, 7, and 9.
2Step 2: Add up the elements on the main diagonal
Add these elements together to get the trace of the matrix. So, the trace of matrix A is \(1 + 7 + 9 = 17\).
Key Concepts
Understanding MatricesIdentifying Diagonal ElementsBasics of Matrix Operations
Understanding Matrices
A matrix is a rectangular arrangement of numbers, symbols, or expressions, arranged in rows and columns. A simple way to visualize a matrix is like a small table that organizes numbers neatly. The overall size or shape of a matrix is described as "m x n," where "m" is the number of rows, and "n" is the number of columns.
An example matrix might look like this:
Understanding matrices is fundamental because they form the foundation for various operations and calculations in linear algebra, such as finding the trace of a matrix.
An example matrix might look like this:
- Matrix A, a 3x3 matrix:
Understanding matrices is fundamental because they form the foundation for various operations and calculations in linear algebra, such as finding the trace of a matrix.
Identifying Diagonal Elements
Diagonal elements are specific numbers in a square matrix (where the number of rows equals the number of columns) that form a diagonal line from the upper left to the bottom right corner. This diagonal is often referred to as the "main diagonal."
Recognizing diagonal elements is crucial for certain calculations, such as determining the trace of a matrix, which is simply the sum of these diagonal values.
- In matrix A, the main diagonal consists of the numbers 1, 7, and 9.
Recognizing diagonal elements is crucial for certain calculations, such as determining the trace of a matrix, which is simply the sum of these diagonal values.
Basics of Matrix Operations
Matrix operations are actions you can perform on matrices, such as addition, subtraction, and multiplication. These operations are essential for solving mathematical problems involving matrices.
One specific matrix operation is calculating the trace. The trace of a square matrix is the sum of all its diagonal elements. It is a straightforward calculation involving adding numbers:
Understanding these operations also helps when exploring more advanced topics like determinants and eigenvalues, which build upon basic matrix concepts.
One specific matrix operation is calculating the trace. The trace of a square matrix is the sum of all its diagonal elements. It is a straightforward calculation involving adding numbers:
- For matrix A, trace is calculated as \(1 + 7 + 9 = 17\)
Understanding these operations also helps when exploring more advanced topics like determinants and eigenvalues, which build upon basic matrix concepts.
Other exercises in this chapter
Problem 151
Find a \(3 \times 4\) matrix \(A\) given that \(a_{i j}=\frac{(i+j)^{2}}{2}\).
View solution Problem 152
Find a matrix \(A_{2 \times 3}\) given that \(a_{i j}=\left[\frac{i}{j}\right]\), where [] denotes greatest integer function.
View solution Problem 154
Find the value of \(x, y, z\) and \(a\) which satisfy the matrix equation \(\left[\begin{array}{cc}x+3 & 2 y+x \\ z-1 & 4 a-6\end{array}\right]=\left[\begin{arr
View solution Problem 155
Find \(x\) and \(y\) if \(\left[\begin{array}{cc}x+y & 2 \\ 1 & x-y\end{array}\right]=\left[\begin{array}{cc}3 & 2 \\ 1 & 7\end{array}\right]\)
View solution